Principles of viscous sintering in amorphous powders: A critical review

Accepted Manuscript Title: Principles of Viscous Sintering in Amorphous Powders: A Critical Review Authors: Mohammadmahdi Kamyabi, Rahmat Sotudeh-Gharebagh, Reza Zarghami, Khashayar Saleh PII: DOI: Reference:

S0263-8762(17)30329-5 http://dx.doi.org/doi:10.1016/j.cherd.2017.06.009 CHERD 2712

To appear in: Received date: Revised date: Accepted date:

16-2-2017 6-6-2017 8-6-2017

Please cite this article as: Kamyabi, Mohammadmahdi, Sotudeh-Gharebagh, Rahmat, Zarghami, Reza, Saleh, Khashayar, Principles of Viscous Sintering in Amorphous Powders: A Critical Review.Chemical Engineering Research and Design http://dx.doi.org/10.1016/j.cherd.2017.06.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Principles of Viscous Sintering in Amorphous Powders: A Critical Review

Mohammadmahdi Kamyabi1,2, Rahmat Sotudeh-Gharebagh1*, Reza Zarghami1, Khashayar Saleh2* Multiphase Systems Research Lab., School of Chemical Engineering, College of Engineering, University of Tehran, Tehran 11155/4563, Iran 2 Sorbonne Universités/Université de Technologie de Compiègne, EA 4297 Transformations Intégrées de la Matière renouvelable, France 1

* Corresponding authors. Tel: +98 21 6697 6851, +33 3 4423 5274

Email addresses: [email protected], [email protected]

Graphical abstract

Caking

x/R = f(viscosity, surface tension, time, relaxation time, compliance, temperature, etc.)

Modeling S/S0 , x/R= f(number of particles, coordination number, etc.)

Densification, porosity, particlesize distribution , cake strength , etc.

1

F = f(x/R)

Highlights:    

Basics in theory of caking in amorphous powders is reviewed Cake formation by the mechanism of viscous flow sintering is investigated Modeling approaches along with analytical and numerical solutions for describing caking in amorphous materials are summarized The considerations are conducted on the multi scales of the phenomenon

Abstract: The present work deals with the main mechanisms involved in caking of amorphous powders and its theoretical background. After a brief outline of the general basics, the review continues with summarizing available models and their analytical solutions predicting interparticle bridge formation as the micro scale description of caking. Outputs of numerical techniques including conventional computational fluid dynamics (CFD) methods alongside with some newer methods such as lattice Boltzmann method (LBM), on capturing the interparticle bridge interface when two particles are sticking together are also investigated. With an overview on the applications of numerical methods for describing multi-particle caking as an intermediate scale, the review continues through investigation of simulations carried out for macro scale caking. The macro scale study surveys numerical methods belonging to the discrete viewpoint -discrete element method (DEM) particularly - while continuum point of view was out of purpose. This review summarizes the basics in caking of amorphous powders along with the modeling approaches predicting viscous sintering mechanism and the use of analytical and numerical methods provided to solve such models. The review characterizes competencies and limitations of the listed methods. Prospects for future research in this area are also highlighted. The result of this study will be useful in identifying the future research directions needed in order to understand leading mechanisms in caking of amorphous powders and also to facilitate the development of more accurate and sophisticated models to predict the kinetics of this phenomenon in process industries. Keywords: Amorphous caking, multi-scale modeling, viscous sintering, CFD, DEM

Nomenclature 2

Latin: 𝑎 B 𝐶 Ca 𝐶1 D d E 𝐹𝑡 Fi,j Fn,ij

Contact area which is equal to 𝜋𝑥 2 , m2 Constant in WLF equation, K Compliance of the material, Pa-1 Capillary number, dimensionless Material’s constant in compliance power law equation Constant in WLF equation, dimensionless Inter-particle distance, m Young’s module, Pa External force on the particles, N Total force between particles i and j, N Normal force between particles i and j in contact, N

𝐼𝑖

Rotational inertia of particle 𝑖, Kg.m2 Fitting constant in Gordon and Taylor equation Fitting constant in Kirchhof’s model Circumference of the contact which is equal to 2𝜋𝑥, m Mass of particle 𝑖, Kg Material’s constant in Eq.10 Average coordination number of particles in aggregate Average pressure on the contact plane, Pa Initial radius of the particles, m Initial radius of the first particle in a two-particle system, m Initial radius of the second particle in a two-particle system, m Radius of particles at time t after contact, m Reynolds number, dimensionless Modified Reynolds number, dimensionless Distance between particles i and j, m Total surface of aggregate, m2 Total initial surface of aggregate, m2 Suratman number, dimensionless Time, s Characteristic time distinguishing zipping stage from adhesive contact stage, s Characteristic time distinguishing stretching stage from zipping stage, s Characteristic time of viscous sintering, s Temperature, K Total torque between particles i and j, N.m Glass transition temperature, K Glass transition temperature of substance, K Glass transition temperature of water, K

𝑘 K0 𝑙 𝑚𝑖 m ̅̅ 𝑁̅̅𝑘

𝑃̅ R0 R1,0 R2,0 R Re Rec rij S S0 Su t t0 tvis tvisc 𝑇 Ti,j Tg Tgs Tgw

3

V Vi 𝑥 𝑥0 𝑥𝑓

Total volume of the particles, m3 Volume of particle i, m3 Radius of the bridge, m Initial radius of the bridge, m Final radius of the bridge, m

δ δc 𝜀̇ θ λ µ µg

Tensile strength, Pa Inter-particle cut-off distance, m Strain rate, 1/s Angle of intersection, rad Relaxation time, s Viscosity (above Tg) , Pa.s Viscosity at Tg , Pa.s

𝜈

Poisson ratio, dimensionless Surface tension coefficient, N/m Angular velocity of particle 𝑖, rad/s

Greek:

σ 𝜔𝑖

1- Introduction: Caking refers to an undesired process where spontaneous agglomeration of individual particles forms coherent blocks of solids called cakes or lumps (Afrassiabian et al., 2016; Christakis et al., 2006; Thakur et al., 2014). Powder caking arises from inter-particle adhesiveness of individual grains which grow in size and change in form. Caking has adverse and irreversible effects on the powder quality and could deteriorate severely the powder properties. Caked materials have generally poor flowability and end-use properties. Therefore, caking is an annoying problem in powder industries such as foods (dairy based (CHUY and LABUZA, 1994; Fitzpatrick et al., 2005; Özkan et al., 2002; Paterson et al., 2005) and others (Cheigh et al., 2011; Christakis et al., 2006; Nijdam et al., 2008; Saragoni et al., 2007)), polymers (KONG et al., 2008; Salman, 1989), fertilizers (Hansen et al., 1998; Rutland, 1991; Silverberg et al., 1958; Walker et al., 1997; Walker et al., 1998), bio-industries (Aguilera et al., 1993; Ganesan et al., 2008a) and pharmaceuticals (Du and Liu, 2008; Provent et al., 1993; Shah et al., 2008). The main mechanisms of caking, the caking rate (kinetics) and the mechanical properties of caked materials vary according to the physico-chemical properties of the material, particles size (and size distribution) and shape along with environmental and storage conditions. Caking results from individual particles adhesion (micro scale) although its visible effects appear only in the macro scale within the bulk material. While densification/shrinkage is the frequently reported 4

parameter indicating the progress of caking in large scales, some other parameters such as caking index (first introduced by Aguilera et al. (Aguilera et al., 1995)) and flowability index (first introduced by Jenike (Jenike, 1964)) were also mentioned as the quantitative characteristics of macro scale caking. Therefore, caking is a complex and multi-scale processes the description of which needs a good understanding of leading mechanisms involved. For several years, researchers attempted to understand the conditions which lead to cake formation-destruction, to predict its rate (Cleaver et al., 2004; Langlet et al., 2013; Özkan et al., 2002; Saragoni et al., 2007), to determine the cake characteristics (Billings et al., 2006; Cleaver et al., 2004; Özkan et al., 2002; Prime et al., 2011), to find ways to prevent caking (Bhandari et al., 1993; Bhandari et al., 1997; Chen and Chou, 1993; Hayashi, 1989; Karatas and Esin, 1990; Lazar et al., 1956; Mehos and Clement, 2008; Peleg and Hollenbach, 1984; Rutland, 1991) and to explore techniques to predict its effects on the flowability of the powder (Fitzpatrick et al., 2005; Fitzpatrick et al., 2007; Mathlouthi and Roge, 2003). Although the experimentation remains the most usual approach for studying caking, modeling was also under attention because of its profits in comparison with experiments. Modeling of caking provides a phenomenological view of this process. Using modeling rather than or in associated with experiments gives the benefits of cost reduction and time saving because a real caking process may occur during several days (for instance see (Langlet et al., 2013)), even weeks or months (see (Cleaver et al., 2004)(. In addition, modeling allows dividing the study of the process into different length and time scales which makes it possible to investigate the system in more details. This advantage becomes dramatically highlighted considering high costs related to accurate micro/mesoscopic experiments on the caking (Sprittles and Shikhmurzaev, 2012). Moreover, when a theory predicts well in certain conditions it can be applied in other situations by considering a series of measurements. This also provides the researchers with the opportunity to predict the behavior of the caking phenomena in a wide range of the materials and conditions and large handling scale. Despite the increasing progress in modeling of the caking, the lack of the resources which have summarized these findings on viscous caking -as the dominant mechanism of caking in amorphous powders- is sensible. This is while caking in amorphous powders have been reviewed very well from the experimental point of view (see (Adhikari et al., 2001; Aguilera et al., 1995; Boonyai et al., 2004; Foster, 2002)). Therefore, this review paper emphasizes the findings in modeling and consequently simulation of the viscous sintering leading to cake formation. Attempting to extract the main contribution of each work is the goal; therefore looking for the details of the methods and exploring the description of the side related issues are delegated to the reader.

2- Background Caking of amorphous powders occurs through sequential phases (see Figure 1). In the early phase of the caking (which is usually named “stickiness”), the powder becomes sticky and the 5

particles adhere to each other. Bridges are formed between particles which tend to reduce the powder flowability. With progress in caking, a stable powder cake is formed. In a later stage of caking, the open pores disappear which leads to the powder shrinkage and therefore an increase in its bulk density (Foster, 2002; Hartmann and Palzer, 2011). Although adhesion may occur through van der Waals forces, the formation of the bridge is a key parameter in most caking cases. However, there are different types of caking bridges including liquid, solid and amorphous bridges. Figure 1

Evaluating phases discussed, it is obvious that caking occurs through a gradual sequence starting from particle-particle coalescence, passing through multi-particle aggregates and ending in large scale shrinkage. Although all these phases are happening simultaneously during caking -maybe at different positions within the powder- the sequence must be completed if observable cake is seen. Therefore, caking is the outcome of a completeness of coalescence of powder in particle level as micro scale, multi-particle level as meso scale and bulk level as large scale. This makes the process of caking as a multi-scale phenomenon. In the following, mechanisms of adhesion for amorphous powders are discussed first and effective parameters in amorphous caking are introduced later.

2-1- Inter-particle adhesion mechanisms Physico-chemical properties of the material (e.g. crystalline/amorphous, water solubility, surface tension, viscosity, etc.) along with humidity and temperature are the primary parameters determining the main mechanisms of adhesion. According to Hartmann et al. (Hartmann and Palzer, 2011), for water-soluble amorphous particles, four following mechanisms of adhesion are possible:    

Plastic/viscoelastic deformation due to van der Waals forces Liquid bridges with moderate viscosity Amorphous bridge with high viscosity Solid bridge due to crystallization

The present review focuses on the formation of amorphous bridge since it is the main mechanism of adhesion leading to caking in amorphous powders. Since the attraction is between similar substances it is better to refer to amorphous bridge as a mechanism of cohesion. Therefore, in the rest of the manuscript the term “adhesion” will be used when general meaning of inter-particle forces are of purpose and the term “cohesion" will be used in the places with special intent of amorphous bridge. Amorphous bridge in the literature was usually classified in solid bridge category. Therefore, when the solid bridge is mentioned, one should note the kind of bridges intended. Crystallization, sintering, partial surface melting and chemical reaction have been

6

categorized under mechanisms of formation of solid bridge by Rumpf for the first time (Adhikari et al., 2001; Rumpf, 1958; Specht, 2006). More recently, the glass transition has been also introduced as a separate potential mechanism for the formation of the solid/amorphous bridge (Farber et al., 2003; Noel et al., 1990). The name of this mechanism comes from a particular property of amorphous substances where the glassy materials become rubbery at temperatures above their glass transition temperature (Tg) so they can mobilize as a flow with high viscosity (Aguilera et al., 1995). Based on this definition, it is more acceptable to categorize glass transition mechanism under the mechanisms relevant to the formation of amorphous bridge rather than solid bridge. The glass transition temperature is the property of amorphous material and depends on molecular morphology and water content of the material where the water acts as a plasticizer reducing Tg (Sperling, 2005). However, above Tg and due to more molecular mobility, the viscosity decreases and the material becomes deformable and can flow under certain internal or external forces. That is why glass transition mechanism is usually called as “viscous sintering”. During the viscous sintering, the flow of amorphous material inside particles leads to the formation of a dynamic neck or bridge path between adjacent particles as Figure 2 shows for a real case. Surface tension acts as a driving force in this dynamics due to large curvature of the surface around the contact area. The bridge growth continues as long as the particles are completely unified and total free surface becomes minimum, unless some external forces stopped the growth. For example, if the environment temperature falls again under Tg, the forming bridge will become solid and nondeformable before complete merging of particles. Although decreased viscosity enables bridge formation, it is worth knowing that the viscosity remains still very high, generally in the order of 106 - 109 Pa.s (Downton et al., 1982; Martínez‐ Herrera and Derby, 1994). Surprisingly there has been much less attention in the literature to the viscous flow sintering in comparison with other mechanisms especially diffusion based sintering common for metal powders (Ross et al., 1981). Figure 2

2-2- Effective parameters in amorphous caking Temperature and relative humidity appear as the most effective environmental parameters of amorphous powders caking (Haider et al., 2014). For this reason, caking is a major challenge in areas with warm and humid weather (Fitzpatrick et al., 2007; Haider et al., 2014). The effects of these parameters directly reflect in Tg where positive T-Tg is prerequisite for probability of observable caking in free flowing glassy powders. The kinetics of caking in amorphous powders is directly related to T-Tg (Foster et al., 2006; Hartmann and Palzer, 2011). At rubbery state (positive T-Tg), the viscosity of the substance decreases with increasing T-Tg which leads to an easier flowing and faster bridge formation. Equation of Williams, Landel, and Ferry (WLF)

7

(Williams et al., 1955) is one of the most referred relations showing dependency of viscosity on T-Tg.

log (

−𝐷. (𝑇 − 𝑇𝑔 ) 𝜇 )= 𝜇𝑔 𝐵 + (𝑇 − 𝑇𝑔 )

(1)

where D = 17.44 (dimensionless) and B = 51.6 K were found suitable for the most polymers (Williams et al., 1955). Besides the increase of temperature (T), a decrease in Tg is another way which accelerates the caking. Most of the amorphous powders are hygroscopic since they absorb water from humid environment (Ganesan et al., 2008b) without dissolving (unlike crystalline materials) (Hartmann and Palzer, 2011). The absorption continues until the thermodynamic equilibrium is reached where water activity in the material equals to the relative humidity (Haider et al., 2014). Because of the small dimension of water molecules, both the viscosity and the elasticity of the amorphous substance decrease with their water content. This behavior is known as the plasticizing effect of water (Haider et al., 2014; Hartmann and Palzer, 2011). Increasing the water content leads to a decrease in Tg approximately with the rate of ~ 10 oC per 0.01g of water/g of material (Hancock and Zografi, 1994; Slade et al., 1989). For instance, Fitzpatrick et al. (Fitzpatrick et al., 2007) brought evidences showing that absorbing only 5% water by a water-soluble amorphous material (lactose in this example) may decrease its glass transition temperature from very high amounts (90 oC) to room temperature. Decreasing in Tg, allows caking to be possible at lower temperatures but in humid conditions. There are also relations for predicting Tg as a function of water content. Popular equation of Gordon and Taylor (Gordon and Taylor, 1952) which was basically adopted for predicting Tg of copolymers can be used (Haider et al., 2014):

𝑇𝑔 (𝑤) =

(1 − 𝑤). 𝑇𝑔𝑠 + 𝑤. 𝑘. 𝑇𝑔𝑤 (1 − 𝑤) + 𝑤. 𝑘

(2)

where Tgw = -135 0C is the suggested glass transition temperature of water (Gordon and Taylor, 1952; Palzer, 2007). In Figure 3 a phase diagram for lactose-water two phase system is shown as an example which shows the effects of water on glass transition of lactose along with solubility and melting curves of the mixture. It is seen that Gordon and Taylor equation with k = 7.3 could predict lactose glass transition very accurately. Isoviscosity curves are also presented based on WLF equation to indicate the caking zone for the material. As it is observable, caking occurs at conditions with relatively low amounts of absorbed water and high temperatures (above Tg). With increasing water content (means decreasing lactose fraction on the graph), caking happens at lower temperatures. Knowing caking zone borders for each material is essential in order to find proper conditions for its storage.

8

Figure 3

Now with understanding the nature of the caking and its effective parameters, the review provides through modeling the caking firstly indicating the challenges ahead of modeling and subsequently simulation of cake formation.

3- Challenges in Modeling In the caking process, since time and length scales of inter-particle adhesion phenomenon are very small, the experimental studies become extremely tedious. Using high quality cameras for capturing bridge growth has become possible only in the recent years (for instance see (Aarts et al., 2005; Paulsen et al., 2012; Thoroddsen et al., 2005)). However, enormous costs related to high accuracy experiments give a strong motivation for developing theoretical models for caking (Sprittles and Shikhmurzaev, 2012). Therefore, it is now clear why describing the caking phenomenon through mathematical formulations (modeling) is of high interests among researchers for several years (Brendel et al., 2006; Chen and Wang, 2006; Cosgrove et al., 1976; Frenkel, 1946; Garabedian and Helble, 2001; Leaper et al., 2002; Weuster et al., 2013). Solving these formulations analytically or numerically was their choice to reach a better understanding of the caking. Although great progresses have been reported on the modeling of the caking, there are still several challenging issues. In the following, these challenges are discussed. One of these challenges is the complex nature of caking where several mechanisms may occur simultaneously in each stages of this multi-physics event (Pan, 2003; Seville et al., 2000; Thakur et al., 2014). Experimental results give an integrated description of the caking in general way where decomposing the results related into separated mechanisms is not easily possible while theories are developed for specified mechanism. Therefore, some inconsistencies among experimental and modeling results may arise from this source. In addition, the multi-scale nature of the phenomenon induces the existence of different time and length scales during caking. Hence a multi-scale method which has the ability to consider all scales of the caking is needed. Therefore, multi-physics and multi-scale nature of the caking constitutes the first challenge in its comprehensive modeling. The second challenge refers to the unavoidable singularity in the dynamics of micro scale caking. Exactly at the time when the particles touch each other and the contact area approaches to zero, the curvature of the interface is infinite. Therefore, the Young-Laplace equation which controls the expansion of amorphous bridge diverges at t=0 after the contact of the particles. This means existing of nonphysical singularity in the dynamic of the problem (Eggers et al., 1999; Sprittles and Shikhmurzaev, 2012). Treating this non-integrable singular point varies in different methods makes the solutions highly dependent on initial conditions. Neither experiments (for the devices accuracy limitations) nor simulations (because of mesh resolution limitations) can extract the information on the absolute contact time. However, all studies start their simulations

9

from a self-defined initial condition, which is usually obtained from experiments or predictions of some static models. If the coalescence is considered as a single phase flow, the known difficulty in simulation of free surfaces is another challenging issue. Stable algorithms are required to continuously adjust the mesh properties according to the surface deformations (Jagota and Dawson, 1990). Therefore, an accurate moving boundary treatment is needed. The solution is not much better in two-phase condition if the coalescence is considered as a two-phase flow. In addition to further computational efforts for two-phase flow systems, serious challenge would be the divergence seen in simulations with high viscosity or density ratios between the phases. This is of importance because the particles exposed to caking conditions are very viscous and dense while the surrounding phase (which is usually the air) is inviscid with low density. For a better understanding of this challenge see (Lörstad and Fuchs, 2004; Selvam et al., 2007). Computational cost is another issue which imposes limitations on numerical methods on the whole scale spectrum. Knowing the problem dependent on grid size (Sprittles and Shikhmurzaev, 2012), mesh fine resolution is required for more accurate implementation of initial contact. However, reaching the singular point at the first moment of contact is impossible. Very large number of particles required for simulation of real operation confirms the inherent limitations of numerical methods in large scale as well. Despite of these challenges in modeling of caking, one cannot ignore its necessity. Benefits of modeling mentioned before provide strong motivations to develop mathematical models for deep understanding of caking with more details. The daily introduction of more powerful computers draws a bright way in front of numerical simulations. In addition, there are fundamental attempts for solving the challenges in simulations of non-physical singularities, free surface problems and high viscosity/density ratios between the phases (Eggers et al., 1999; Lörstad and Fuchs, 2004; Selvam et al., 2007; Sprittles and Shikhmurzaev, 2012).

4- Two-Particle Coalescence As mentioned earlier, inter-particle adhesion is the first stage of caking and describes caking in micro scale. When two amorphous particles are brought into contact, cohesive forces make irreversible joints. Since these inter-particle joints appear in the particle scale, they belong to the micro level description of the caking. In this section, the status of the viscous sintering in twoparticle coalescence general category will be determined and modeling attempts in the micro scale cohesion related to the caking is reviewed with the focus on viscous sintering. Although fluid flow which is the criterion for viscous sintering mechanism is mostly considered as Newtonian in the literature it may shows non-Newtonian effects e.g viscoelasticity which is common for amorphous materials. Therefore, in following sections, first Newtonian assumption is discussed, and then elastic and viscoelastic effects are reviewed.

10

4-1- Newtonian viscous coalescence While particle coalescence is the essential stage in caking process, it also occurs in many other physical situations. Examples are the water drop formation leading to rain (Kovetz and Olund, 1969), spray cooling (Grissom and Wierum, 1981), emulsion stability (Dreher et al., 1999) and 3D-printing applications (Derby, 2010; Singh et al., 2010). In other words, caking in micro scale is a member of a bigger class of particles coalescence phenomenon. Therefore, similarities among these applications and micro scale description of caking may be expected. In particle coalescence in general, after a contact between two particles is created, an interparticle bridge forms (see Figure 2) and grows with time under the effects of capillary force where viscous and inertial forces resist against its growing (Eggers et al., 1999; Paulsen et al., 2011; Paulsen et al., 2012). During this physical process, mass, momentum and energy are conserved. Neglecting energy balance (isothermal conditions) along with assuming Newtonian flow of the material, continuity and Navier-Stokes equations were usually supposed to govern on the micro scale viscous sintering. If the surrounding phase is neglected, the capillary force acts as the boundary condition through applying Young-Laplace equation. If the surrounding phase is also considered -although it was rarely considered in the literature- the surface tension force appears in the Navier-Stokes equations as a source term. In viscous sintering of a two-particle system, gravity and inertia are negligible in comparison with viscous and capillary forces because of very small length scale and high amounts of viscosities (Djohari et al., 2009; Garabedian and Helble, 2001). However, for achieving analytical solutions, usually the energy conservation along with mass balance were solved with simplifications. Simplifying assumptions were also used instead of direct solution of momentum conservation equation. Several researchers attempted to model the formation of the inter-particle bridge derived by viscous flow. First attempt to model the coalescence of pair particles was reported by Frenkel (Frenkel, 1946) in which an energy conservation model was proposed to formulate the bridge growth with time. In his model, a two-particle system with the same radii of the particles was considered. It was supposed that the shape and radiuses of the particles remain constant during the coalescence and the process continues with a constant strain rate within an elongational flow. Based on first law of thermodynamic (conservation of energy), it was supposed that the energy released by reducing the free surface of the system during the coalescence was totally consumed as the work of resistant viscous force (Ristić and Milosević, 2006) as follow: −

𝑑𝐴𝑖 = ∭ 3𝜇𝜀̇𝑖2 𝑑𝑉𝑖 𝑑𝑡

(3)

The left hand side of the equation describes the energy released during surface retraction whereas the right hand side of the equation represents work of resistant viscous force. The assumption of constant strain rate (𝜀̇) makes Frenkel’s model needless of solution of momentum conservation. Supposing constant radius for the particles violates the conservation of mass (Tarafdar and 11

Bergman, 2002).This issue is shown in the Figure 4 where the gray area indicates schematically the excess mass exerted by Frenkel’s model. Based on this figure, R0 is the initial radius of the particles, and R, x and θ which are radius of the particles, and radius and angle of intersection respectively are changing during coalescence. Kinetics of coalescence determines how fast are these changes. Figure 4

Analytical solution of Frenkel’s model led to the following law for the growth rate of the bridge (see Figure 4): 𝑥 𝑥 3𝑡 = =√ 𝑅 𝑅0 2𝜇𝑅0

(4)

Rumpf’s model (Rumpf, 1958) also led to the same power law model with small variations. Because of the assumption of small dihedral angle (θ), both of these models are valid only at early stages of coalescence when the particles have not been deformed considerably (Hartmann and Palzer, 2011). Eshelby modified Frenkel’s model by considering mass conservation (which is equal to the volume conservation for incompressible flow) (Shaler, 1949). Volume conservation dictates that the total volume, V, of the two particles of volumes V1 and V2 remains constant during the coalescence, so: 𝑑𝑉 𝑑𝑉1 𝑑𝑉2 = + =0 𝑑𝑡 𝑑𝑡 𝑑𝑡

(5)

This model is known as Frenkel-Eshelby model and led to the following rate law: 𝑥 𝑡 =√ 𝑅 𝜇𝑅0

(6)

Obviously in this model R is changing and is not the same as R0. The progress of such single-equation models for all two-particle coalescences was continued until Eggers et al. (Eggers et al., 1999) revealed that there are more than one regimes in capillary driven coalescence of two-particle systems; therefore single-equation models are not proper for describing two-particle coalescences for all ranges of the materials. This matter helped to distinguish between viscous sintering and other surface-tension-driven flow coalescences.

4-2- Newtonian coalescence regimes

12

As mentioned before, the bridge growth is competitive among the surface tension force which is the driving of the coalescence and viscous and inertial forces which are resistant forces. The viscous force is due to friction between the layers of the coalescing medium and is characterized by viscosity whereas the inertial force is due to convective momentum of the fluid and is represented by density. These forces terms are available in Navier-Stokes equations as below: 𝜕𝑉 𝜌 ( + 𝑉. ∇𝑉) = ⏟ 𝜕𝑡 𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝐹𝑜𝑟𝑐𝑒𝑠

−∇𝑃 ⏟

+⏟ ∇. (µ(∇𝑉) + (∇𝑉)𝑇 ) +

𝐹 ⏟

𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐹𝑜𝑟𝑐𝑒𝑠

𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝐹𝑜𝑟𝑐𝑒𝑠

𝐸𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝐹𝑜𝑟𝑐𝑒𝑠

(7)

Eggers et al. (Eggers et al., 1999) suggested that generally two distinct regimes are observable during two-particle coalescence based on relative values of surface tension, viscosity and density of the particles. These two regimes were referred to viscosity dominant and inertia dominant regimes (Aarts et al., 2005; Sprittles and Shikhmurzaev, 2012). If viscous force is dominant so that inertial force is negligible (i.e. high viscosity or low density), the coalescence takes place in viscous dominant regime. At contrary situations (i.e. high density or low viscosity), the regime of coalescence is inertia dominant. Based on this classification, viscous sintering which is the main mechanism of bridge formation in amorphous powders is a viscosity dominant process because of high viscosity of the materials involved. Since Stokes (Navier-Stokes equations without inertia terms) and Euler (Navier-Stokes equations without viscous terms) equations are assumed to be governing on dynamic of coalescence in viscous dominant and inertia dominant regimes respectively, these regimes are also called as Stokesian and Eulerian regimes (Sprittles and Shikhmurzaev, 2012). Figure 5 shows these regimes classification along with examples of their related physical phenomena. Figure 5

Table 1 presents the characteristic parameters of the flow at two different regimes. Ca is capillary number and We is Weber number. Table 1

Reynolds number which indicates the proportion of inertial and viscous forces determines the coalescence regime. For capillary driven coalescence, a modified Reynolds is defined based on capillary velocity to include surface tension force as well (Aarts et al., 2005; Sprittles and Shikhmurzaev, 2012): 𝑅𝑒𝑐 =

𝜌𝑈𝑣𝑖𝑠𝑐 𝑅 𝜌𝜎𝑅 𝑅𝑒 2 𝑅𝑒 = 2 = = µ µ 𝑊𝑒 𝐶𝑎

(8)

This definition of Reynolds number can be also defined in term of Suratman (Su) number (Martínez‐ Herrera and Derby, 1994). Ohnesorge number (oh) which is more applied in printing applications (McKinley and Renardy, 2011) is also another definition of the same concept: 13

(9) 1 √𝑊𝑒 = 𝑅𝑒 √𝑅𝑒𝑐 Aarts et al. (Aarts et al., 2005) suggested 𝑅𝑒 = 1 as the border of two regimes where 𝑅𝑒 < 1 shows viscosity dominant and 𝑅𝑒 > 1 indicates inertia dominant regimes. By the same logic, and considering the local Reynolds number ( Rec,x = ρσx/µ2 ) Eggers et al. (Eggers et al., 1999) concluded that the initial time of any two-particle coalescence is under Stokesian regime because x has very low values at early times after contact. However, this argument was later contradicted by Paulsen et al. (Paulsen, 2013; Paulsen et al., 2012) showing that all coalescence processes are initially governed by an inertia resistance regime. They revealed that a third regime of inertiallylimited viscous (ILV) regime also exist in addition to the previously introduced two regimes. The phase diagram which they suggested is the newest characterization of the coalescence regimes of its kind. Based on this diagram shown in Figure 6 all particle/droplet coalescences are in an inexorable inertia regime in their early times after contact. This fact is notoriously in contrast with the consequence of Eggers et al. (Eggers et al., 1999). Therefore, although it was frequently considered as an obvious fact, attributing viscous dominant to the all stages of viscous coalescence may not be true. 𝑂ℎ =

Figure 6

It is frequently demonstrated in the literature that the bridge grows as x2 ∝ t for Eulerian coalescence (Aarts et al., 2005; Duchemin et al., 2003; Thoroddsen et al., 2005) but the results raise some ambiguity in Stokesian regime. Although x ∝ t.ln(t) is suggested by some researchers (Eggers et al., 1999; Hopper, 1984), but the logarithmic part is suspect and eliminated by evidences of some others (Aarts et al., 2005; Yao et al., 2005). Even x2 ∝ t (somehow Frenkel’s model) has been also suggested for Stokesian coalescence (Gross et al., 2013). However, the results reported in the literature are somehow controversial. Part of the reason for these inconsistencies can be explained by the attempts of Paulsen et al. (Paulsen, 2013; Paulsen et al., 2011; Paulsen et al., 2012) where they introduced the third regime of coalescence as mentioned before. Actually, the try to fit a unique model on the experimental data which were or were not (depending on initial conditions supposed) simultaneously from two ILV and Stokesian regimes may lead to differences in proposed correlations. Different imposed initial conditions and common numerical errors of course are the other reasons for inconsistencies among the main conclusions of the works.

4-3- Elasticity and viscoelasticity effects In most models, it was supposed that the bridge growth through viscous flow mechanism is started instantaneously after the contact occurred. However, this assumption may be valid if only the particles are totally rigid before undergoing a viscous flow. However, in reality, when two elastic particles become into contact, even without external load, a non-zero contact area forms because of cohesive forces. The source of such cohesive force is different (e.g. surface energy in

14

particular surface tension). Johnson et al. (Johnson et al., 1971) developed famous JohnsonKendall-Roberts (JKR) theory which predicts radius of such contacts for linear elastic spheres. Considering viscoelastic nature of particles, Lin et al. (Lin et al., 2001) summarized the sintering process of non-rigid particles in three consecutive stages: 1) Elastic adhesive contact (predictable by JKR or similar theories); 2) “Zipping” or viscoelastic cohesive contact growth; 3) “Stretching” or viscous contact growth. The third stage is actually the same as classical viscous flow mechanism which has been discussed in previous sections 4-1 and 4-2. Figure 7 presents a schematic of these three stages with their related characteristic times. Figure 7

For a viscoelastic material, compliance is a substantial parameter. The effective creep compliance of a Maxwell material (as a simple viscoelastic model) can be written in the following form as (Li et al., 2011): 𝐶(𝑡) = 𝐶0 + 𝐶1 𝑡 𝑚

(10)

where 𝐶0 = (1 − 𝜈 2 )/𝐸 is instantaneous compliance, C1 and 0 < m ≤ 1 are material constants. According to Figure 7, t0 which is the characteristic time indicating the boundary between elastic and viscoelastic regimes, is defined as below (Li et al., 2011): 7⁄ 3

8𝑅 9𝜋𝐶0 𝜎 𝑡0 = ( ) 63𝜋 3 𝐶1 𝜎 𝑅

𝑅 ( )2 𝛿𝑐

(11)

where 𝛿𝑐 is the inter-particle cut-off distance at which for 𝑑 > 𝛿𝑐 cohesive force becomes zero. Lin et al. (Lin et al., 2001) suggested that for sufficiently short times (𝑡 < 𝑡0 ), the material is unrelaxed and the elastic adhesive contact JKR theory is appropriate: 𝑥 3 18𝜋𝐶0 𝜎 ( ) =( ) 𝑅 𝑅

(12)

Within elastic stage, the bridge radius is not time dependent and a constant contact area is expected. This matter is also obvious from the Eq. 12. tvis which is a characteristic time presenting the (13) transition time between viscoelastic adhesive contact and viscous flow stages is obtained from the following equation: 𝑡𝑣𝑖𝑠 = 15

𝑅 32𝐶1

( 𝜎

63𝜋 3 2⁄ 𝛿𝑐 4⁄ ) 5( 𝑅 ) 5 2

At 𝑡 > 𝑡0 but still short compared with tvisc, the contact radius for a Maxwell material is predicted from equation 14: 1⁄ 7

𝑥 63𝜋 3 ( )=( ) 𝑅 8

1⁄ 7

𝛿𝑐 ( ) 𝑅

𝐶1 𝜎 1⁄ ( 𝑡) 7 𝑅

(14)

For stretching stage, they reached to an equation which is identical to Frenkel’s relation by considering µ = 1/4C1. Several experimental works detected significant differences between viscoelastic and Newtonian bridge growth (Bellehumeur et al., 1998; Mazur and Plazek, 1994; Scribben et al., 2006). When viscoelastic effects are brought up, stress relaxation and chain inter-diffusion influence on the kinetic of sintering (Mazur and Plazek, 1994). While Bellehumeur et al. (Bellehumeur et al., 1998) demonstrated that the Newtonian model overestimates the growth rate for viscoelastic materials, a reverse outcome is obtained by Mazur and Plazek (Mazur and Plazek, 1994) where they saw that sintering time for poly-tetrafluoroethylene (PTFE) and several acrylic resins was meaningfully shorter than predictions by Newtonian models. The reason for these differences is still unclear and requires further investigations.

4-4- Analytical models After introducing basics in theory of two particles coalescence it is needed to summarize the literature results for the bridge growth predictions. In Table 2, a list of the pioneer modeling and their related analytical solutions for describing bridge growth is presented. This list contains the main features, assumptions/limitations and results of the methods. The resources summarized here belong to particle coalescence in general way or those for Stokesian regime which is dominated in viscous sintering. Hence, the results which were specifically developed for Eulerian regime are not presented in this table. This list is sorted based on historical progress of the solutions. Presented symbols are according to notation in Figure 4. Table 2

First models of two particles sintering were mostly originated from Eq. 3 but with different assumptions. As mentioned before, applicability of all single-equation models for viscous sintering depends on their validity in viscous regime. However, Frenkel’s equation as the first model gave a general view of the problem. Eshelby modified this model to make it consistent with conservation of mass but it was still constrained by assumption of small dihedral angle θ. Hopper ( Hopper, 1990; Hopper, 1984) and Pokluda et al. (Pokluda et al., 1997) used numerical integrations for solving the analytical models in their more general forms.

16

Researchers also developed models for prediction of sintering by considering non-Newtonian nature of the amorphous materials. Lontz (Lontz, 1964) simply replaced an exponentially timedecreasing viscosity to the Frenkel’s model. The model developed by Bellehumeur et al. (Bellehumeur et al., 1998) for UCM viscoelastic materials was able to predict sintering data of two propylene-ethylene copolymers only by considering large values for the relaxation time which were not based on experimental data. The model predictions were not consistent with the results of Mazur and Plazek (Mazur and Plazek, 1994) for PTFE particles. Scribben et al. (Scribben et al., 2006) conjectured that the steady-state assumption may not be true for high Deborah number conditions. Therefore, they presented a model for viscoelastic cases by using UCM constitutive equation in its transient mode. Although viscoelastic materials are wide and can be described with several constitutive equations, only simple UCM models have been investigated for sintering problem. Considering isothermal conditions, and therefore ignoring thermal effects on viscosity of the material is a major neglecting point in all available analytical models. Referring to the discussions in section 2.2 this matter is a very important issue in sintering of amorphous materials. This issue along with simple geometries supposed for the particles are the main missing parameters which can make serious differences between analytical predictions and experimental data. Although analytical approaches usually give the exact solution of any physical phenomenon which is converted to mathematical equations, it should be noted that these solutions are only valid within the specified assumptions and limitations. Some of these assumptions and limitations were reported in Table 2. Therefore, it seems numerical simulations are powerful options to help mathematical modeling for eliminating limitations in order to give more realistic results. However, numerical methods have their own shortcomings as discussed before.

4-5- Numerical solutions Actually conservation of mass, momentum and energy are describing the problem of interparticle coalescence. Energy balance is ignored if isothermal condition is considered. Numerical techniques, in particular CFD methods are needed to solve the series of governing equations on coalescence process. This is because of the usual inability of the analytical methods to solve the governing equations in their nonlinear partial differential format. When two particles are adhering, at least two phases are present: the particles phase and the surrounding phase which is usually the air. However, most of simulations considering the bridge growth neglected the effect of surrounding environment and therefore they treated the bridge formation as a one phase and free surface problem. In this condition, surface tension was imposed to the problem as a boundary condition. However, this assumption is not far from reality when the surrounding material has low density and viscosity in comparison with the density and the viscosity of particles. There are also some simulations in the literature dealing with two-phase mode by evaluating the effects of outer phase (Eggers et al., 1999; Hiram and 17

Nir, 1983) or only by considering the outer phase in order to use benefits of some two-phase numerical methods (Kirchhof et al., 2009). In these cases, surface tension was added to the momentum equation as a source term. Independent of the number of the considered phases, the particle boundary changes with time during the dynamic coalescence. Therefore, a moving interface simulation is of interest. Generally there are two main approaches for CFD simulation of problems with moving interfaces: front (interface)-tracking (or sharp-interface) and front (interface)-capturing methods (Cristini and Tan, 2004; Pino-Muñoz et al., 2014). Front-tracking methods follow the interface on a movable (deformable) mesh while in the front-capturing methods the interface is detected by a phase function discretized on a fixed grid. The straightforward definition of the interface in front tracking approach makes these methods appropriate for treating free surface (as one phase) problems. On the other hand, front capturing methods benefit from lack of need to mesh cut-andconnect operations. Marker-and-cell (MAC) method, boundary integral method, finite element method (FEM) and immersed boundary method are the examples of front-tracking methods (Cristini and Tan, 2004; Pino-Muñoz et al., 2014). Volume of fluid (VOF), level-set, lattice Boltzmann, partial-miscibility-model and phase-field methods belong to front-capturing family methods (Cristini and Tan, 2004; Pino-Muñoz et al., 2014). Although front tracking methods are very powerful in simulating the coalescence phenomenon, they faced with difficulties through its simulation (Cristini et al., 2001; Zhou and Derby, 2001). Failure in simulating coalescence and break up by using sharp interface models has been reported because of the formation of singularities in flow variables (Hou et al., 1997) and complex mesh cut-and-connect algorithms which are necessary to follow the interface changes (Cristini et al., 2001; Tryggvason et al., 2001; Zhou and Derby, 2001). Despite of this fact, these methods were frequently used for numerically investigation of inter-particle coalescence. On the other hand, interface capturing methods are ideal for simulating coalescence (Cristini and Tan, 2004) but they also face with difficulties when the viscosity/density ratios between the phases are high (i.e. caking phenomenon) (Lörstad and Fuchs, 2004; Selvam et al., 2007). After choosing suitable method, another important issue is the initialization of the simulation. In some cases such as Menchaca-Rocha et al. (Menchaca-Rocha et al., 2001) it is clearly mentioned that the results of bridge growth is highly mesh dependent at least in the early times. By increasing the mesh resolution, the radius of the curvature decreases and tends to the required singular initial condition. Escaping initial singularity, others such as Paulsen et al. (Paulsen et al., 2012) directly mentioned that their results were extracted only after the transients from the initial conditions have decayed. The leading numerical researches dealing with coalescence in particle level (micro-scale) are summarized in Table 3 with their main features, limitations and results. Since there are many works simulating coalescence of particles, here relevant works were discussed on viscous sintering mechanism. Ross et al. (Ross et al., 1981) were the first who applied FEM to evaluate 18

the bridge growth rate. Their solution overpredicted the bridge growth rate for intermediate bridge sizes, in comparison with the results of geometry-fixed models (such as Frenkel’s model). However, it was concluded that power-law models with a constant power value (such as 0.5 in Frenkel’s model) are not sufficient to predict the coalescence process correctly. Ross et al. (Ross et al., 1981) suggested the power value varied monotonically with bridge growth. The results of Hiram and Nir (Hiram and Nir, 1983) approved this fact, but they suggested a unique exponential model to fit the results of bridge growth. The results of Hiram and Nir (Hiram and Nir, 1983) approved this fact, but they suggested a unique exponential model to fit the results of bridge growth. More recently, with increasing power of numerical methods, researchers attempted to evaluate micro scale sintering of particles with more complex geometries. For instance, a series of studies were carried out to evaluate the behavior of coalescence of two different sizes or shapes of particles (Van de Vorst, 1994; Van de Vorst et al., 1992; Van de Vorst and Mattheij, 1992). Boundary element method (BEM) was preferred in these works because of less computational efforts needed in comparison with FEM. This is due to fact that in BEM, the calculations are performed on the points on the boundary while the network should be remeshed on all the domain in the FEM. However, almost same parameters were also evaluated with FEM providing more details (Martínez‐ Herrera and Derby, 1994, 1995; Zhou and Derby, 1998, 2001). In addition to front-tracking methods, several studies on micro scale viscous sintering were conducted by using front-capturing methods especially VOF method (Djohari et al., 2009; Kirchhof et al., 2012; Kirchhof et al., 2009; Wang et al., 2010). Using VOF, computational costs increases since meshing and solution are required for the domains of both involved phases. The close agreements were reported between the model and experimental data (Kirchhof et al., 2012; Wang et al., 2010) and comparison between results of viscous sintering and diffusion mechanism was also reported (Djohari et al., 2009). In Recent years, it was also attempted to apply Lattice Boltzmann method (LBM) for simulation of micro scale caking (Gross et al., 2013; Varnik et al., 2013). Although LBM provides a benefit over conventional CFD methods in dealing with moving boundaries but its outcome for sintering problem was not something new and just confirming Frenkel’s model of coalescence. In fact, these studies were shown the applicability of LBM in sintering simulations and future works are highly demanded. Considering the need for large scale simulations with information from smaller scales, some recent studies attempted to extract inter-particle force models from computations on micro scale (Katsura et al., 2015; Wakai et al., 2016). These force models should be applied in large scale simulations to evaluate their applicability. A brief summary of numerical methods applied for simulating viscous sintering in micro scale is presented in Table 3. Table 3

With a glance at Tables 2 and 3, one can see inconsistencies among modeling approaches and their results for analytical and numerical methods. Most analytical solutions are based on approaches for solving mass and energy balances where they have excluded momentum balance 19

by simplified assumptions such as constant strain rate. In these models, it was assumed that all energy released by surface tension force during bridge growth is consumed by viscous dissipation. Therefore, heating of the material by the released energy was neglected. This is while numerical methods were mostly based on solving Navier-Stokes equations and neglected energy balance instead. Therefore, both of these approaches have their own deficiencies and related simplifications, definitely lead to the limitations in model predictions. Several studies demonstrated the importance of considering each of these three conservation laws. Therefore, it seems that there is no way for neglecting any of conservation laws and future works should prefer to consider the problem in its general form. In particular, dependency of the physical properties of the material on the temperature and water content should be considered in the models and different viscoelastic models of amorphous substances should be evaluated.

5- Multi-particle coalescence Estimating the behavior of coalescence of particles when more than two particles are touching each other is of interest as the aggregation phenomenon. Jagota and Dawson (Jagota and Dawson, 1990) observed a localized flow was formed only in the small area near the bridge. Based on this observation, several studies ignored the effects of presence of simultaneous contacts at least at early stages of sintering (Martínez‐ Herrera and Derby, 1995). However, now it is known that several contacts are not independent of each other even at first stages of sintering (Kirchhof et al., 2009). In addition, investigating coagulation of a few connected particles as the meso scale description of caking process is a kind of a link between micro and macro scales of the multi-scale caking. Generally within a multi-scale modeling, detailed interactions obtained from small scale models can be averaged and used for calculating interactions at larger scale (Norouzi et al., 2016; Van der Hoef et al., 2006). Therefore, more accurate results are obtained through multi-scale approach since it is consistent with reality of the phenomena. Actually, multi-scale view of amorphous caking is a missing link between the models and the reality. With an increasing number of particles, analytical solutions are rarely used. However, by extending Pokluda’s model (Pokluda et al., 1997), expressions were proposed for the mass and energy balances of the aggregates (Eggersdorfer et al., 2011). This model combined energy balance with general model of Kadushnikov et al. (Kadushnikov et al., 2001) for the mass balance of ensemble of particles. Apart from this case, other studies on multi-particle sintering systems were mostly based on numerical solutions. Considering periodic boundary conditions on two-particle system made simulation of infinite line of particle to be possible (Martínez‐ Herrera and Derby, 1994; Ross et al., 1981; Tarafdar and Bergman, 2002). Ross et al. (Ross et al., 1981) were the first who simulate viscous sintering of an infinite line of particles by using FEM. They found that the bridge growth rate was faster in comparison with predictions of two-particle fixed-geometry models (such as Frenkel’s model). Jagota and Dawson (Jagota and Dawson, 1988) also simulated infinite line of particles and surprisingly observed that Frenkel’s model overpredicted the bridge growth rate. Martínez‐ Herrera and 20

Derby (Martínez‐ Herrera and Derby, 1994) also found that the densification rate is faster in infinite line of particles in comparison with two-particle case. The same assumptions and solution approach were used for simulating two-particle and many-particle systems in their work; hence the differences between these cases are not related to model assumptions or numerical limitations. It can be concluded that there are intrinsic differences between two-particle and many-particle systems. Numerical solution also gave this opportunity to evaluate sintering problem in non-isothermal conditions (Bergman, 1997; Tarafdar and Bergman, 2002). In these studies a simple onedimensional equation (Pokluda’s model) was combined with heat conduction model to evaluate the compaction of particles. Although the effects of temperature on density and thermal conductivity of particles were considered but still viscosity dependency on temperature- which is an important issue in caking of amorphous materials- was missing. Detailed geometrical dynamic modeling of multi-particle systems was described by solving Navier-Stokes equations numerically (Kirchhof et al., 2009; Martínez‐ Herrera and Derby, 1995; Zhou and Derby, 1998). It was seen that some phenomena such as anisotropic shrinkage and inward rotation of particles can be seen only in 3D simulations as more considered (Zhou and Derby, 1998). Kirchhof et al. (Kirchhof et al., 2009) Investigated sintering of different morphologies of aggregates with the same number of particles. They found that, densification is faster and final equilibrium is reached sooner in more compact morphologies compared to widespread morphologies. Despite great advances in multi-particle sintering simulations, the effect of the number of particles and coordination number on sintering of multi-particle systems has not been determined yet. A brief summary of these studies on multi-particle systems of sintering is represented in Table 4. Describing the aggregation phenomenon with population balance method is a common way instead of solving Navier-Stokes equations. These studies were mostly extending existing analytical models in combination with Smoluchowski coagulation kernel for describing multiparticle cases. Using this approach and by ignoring the spatial variables dependency, more number of adjacent particles were simulated. In other words, using this approach, one can follow the evolutions in aggregate size with time but cannot follow spatial changes (i.e. the real shapes of the aggregates). These features can be followed through CFD methods but with high computational costs as mentioned before. The same issue decreases computational efforts combined with population balance simulations and makes them more implementable for coalescence of more number of particles. Using population balance method, it is needed to develop coagulation kernels representing caking. Therefore, attempting to derive appropriate coagulation kernels for general caking was the pioneer in this topic. Applying these kernels, aggregation evolution caused by viscous sintering was investigated through population balance method (Eggersdorfer et al., 2012; Koch and Friedlander, 1990; Tandon and Rosner, 1999; Tsantilis et al., 2001; Xiong and Pratsinis, 21

1993). Outcome of such solutions usually determined the changes in fractal dimension of the aggregates and some other parameters such as radius of gyration and total surface area of the aggregates. Another thing which was seen by population balance modeling is the effect of collision frequency on coalescence rate.

6- Bulk Scale Modeling Bridge formation which occurs in micro scale is prerequisite for caking but it is not a sufficient criterion. The success in this stage will lead to large scale observable cake only if favorable circumstances exist. These circumstances are mostly environmental conditions such as homogeneous humidity and temperature distributions within bulk material and compliant external driving forces like compression load. Actually, if the bridge growth occurs in abundance and irreversibly in micro scale and continue through meso scale, then finding macro scale effects is expectable. In other words, the rate of kinetic of caking depends on all micro, meso and macro scales rates. There are generally two viewpoints in macroscopic modeling of particulate materials: continuum (Eulerian) and discrete (Lagrangian) viewpoints; which describe the bulk powder’s behavior in according with continuum and discrete phases, respectively. Performing a closed analytical solution for many particles systems is quite difficult and somehow impossible due to their complexity (Varnik et al., 2013). Therefore numerical methods are somehow the only available solution for theoretically understanding caking in large scale.

6-1- Continuum Theory of Sintering Assuming a sintered material as a porous media including a phase of substance and a phase of voids, predicting large scale properties of a caked (sintered) powder became possible through continuum theory of sintering (Olevsky, 1998). These properties are including stress and stain fields, density, grain size fields, and shape/dimension or densification of compact powder (Pan, 2003). According to the theory, powders compact is treating as a continuum solid which is experiencing plastic deformation due to sintering. Second law of thermodynamics in terms of Clausius–Duhem inequality acts as the governing equation on the bulk system and sintering is imposed through constitutive laws. It is worth to mention that the continuum theory of sintering is well established and there are several review papers comprehensively summarized the findings in this area (see (Cocks, 1994; Green et al., 2008; Olevsky, 1998; Pan, 2003)).

6-2- Discrete Methods Following the motion of discrete particles, macro scale modeling of the caking can be made through Lagrangian approach. Naturally due to large number of particles in bulk scale, using numerical approaches for solving the Lagrangian form of the equations is necessary. Li et al. categorized molecular dynamics (MD), Brownian dynamics (BD), dissipative particle dynamics (DPD) and discrete element method (DEM) as the Lagrangian numerical methods for solving 22

particulate flows. The key factor parameter which is common in these methods is inter-particle adhesive force. There are generally different inter-particle forces such as van der Walls, liquid bridge, solid bridge, friction forces and electrostatic forces (Hede, 2006; Seville et al., 2000). The essence of inter-particle forces differs based on length and time scale. For example, only friction forces and forces associated with liquid/solid bridges are significant in systems with particle sizes above 10 nm (Rhodes, 2008). In this way, the applicability of the solution methods is also dependent on length and time scales. Figure 8 represents this dependency. Caking is a concern for particulate flows with the grain size not below micrometer. It is a slow process hence its time scale is also relatively high. Therefore, according to Figure 8, discrete (sometimes called distinct) element method (DEM) is the most relevant method for simulating industrial powders under caking conditions. Moreover, DEM is more convenient as in caking there is no convective flow and particles are in contact. Figure 8

The DEM which first introduced in 1979 by Cundall and Strack (Cundall and Strack, 1979), is a numerical method for simulating the behavior of particulate flows through discrete viewpoint. Considering the individual grains as a set of particles with physical properties, the method involves the particle-particle and particle-wall forces to predict the bulk scale behavior of whole particles based on integration of Newton’s second law. Based on the soft sphere model, a particle may have two types of translational and rotational motions. The equations of motions for individual spherical particle i with the mass mi and the moment of inertia Ii can be written as (Norouzi et al., 2016):

𝑚𝑖

𝑑𝑽𝑖 𝑑 2 𝒓𝑖 = 𝑚𝑖 2 = ∑ 𝐹𝑐𝑜𝑛𝑡,𝑖,𝑗 + ∑ 𝐹𝑛𝑜𝑛−𝑐𝑜𝑛𝑡,𝑖,𝑗 + 𝑚𝑖 𝒈 𝑑𝑡 𝑑𝑡 𝑗

𝐼𝑖

𝑑𝜔𝑖 = ∑ 𝑇𝑖𝑗 𝑑𝑡

(15)

𝑗

(16)

𝑗

where Tij = (-1/2) rij × Fij. Inter-particle forces can be either contact or non-contact which means respectively the forces act only when a physical touch between two particles is created and the forces which act remotely. Since sintering cohesive forces act naturally after a contact occurred, free sintering simulations can be performed by considering only contact forces. Despite of best model parameters fittings, failure in predicting polymer particles behavior is reported in the DEM simulations which didn’t consider such inter-particle forces (Moysey and Thompson, 2007). Therefore, implementing DEM along with considering cohesive forces is accountable in simulating caking of polymers. 23

Generally three approaches are available in the literature for implementing adhesive forces with DEM method (Guo and Curtis, 2015). Simpler approach is using non-physical general lumped models independent of contact mechanisms or sources. For instance Grima & Wypych (Grima and Wypych, 2011) experimentally calibrated the multiplication of a constant cohesion energy density and the contact area as the cohesive force. Constant-intensity normal force was also applied (Alexander et al., 2006) (for example see Weber and Hrenya (Weber and Hrenya, 2006) and Brewster et al. (Brewster et al., 2009)). Another approach which is more common in geomechanics is describing contact adhesive forces based on mechanical model parameters such as cohesive strength constant and inter-particle friction angle (Jiang et al., 2011; Utili and Nova, 2008; Zhang and Li, 2006). For validation of these kinds of adhesive models, accurate experimental measurements are needed. Therefore, these two approaches are general and independent of the mechanism of the caking. Only more advanced DEM simulations (as the third approach) are using interparticle force models which are coming from micro/meso scale investigations and are specialized according to the mechanism of the coalescence (Guo and Curtis, 2015). Actually multi-scale modeling would be completed by using these kind of simulations. Although DEM wide applications have been shown through simulating sintering via diffusion mechanism in absence (Henrich et al., 2007; Martin and Bordia, 2009; Martin et al., 2006; Martin et al., 2014; Martin et al., 2015; Parhami and McMeeking, 1998; Wonisch et al., 2007; Wonisch et al., 2009) or presence of fluid flow (Kuwagi et al., 2000), the method recently showed its ability in simulation of viscous mechanism of caking (Luding et al., 2005; Mansourpour et al., 2014a, b). There are several models for describing normal and tangential forces between particles but most of them are specialized for sintering induced by diffusion mechanism. For instance see (De Jonghe and Rahaman, 1988; Martin and Bordia, 2009; Parhami and McMeeking, 1998; Riedel et al., 1994; Tsuji et al., 1992) for normal force models and (Raj and Ashby, 1971; Riedel et al., 1994) for tangential force models. There are only limited DEM studies which specifically simulated caking induced by viscous flow mechanism and most others referred to only simple DEM models. The consequence of this lack of knowledge is a deep gap between micro/meso and macro scales viscous sintering simulations where a multi-scale view of the problem has not been obtained yet. However, in the following the general common force models are explained which were used in viscous DEM sintering simulations at least in one research. There are different approaches for implementing inter-particle adhesive forces in DEM simulations. Along with using a constant force (Thakur et al., 2014) a simple approach which is common and independent on the type of the mechanism, is the suggestion of Kuwagi et al. (Kuwagi et al., 2000) based on Kuczynski’s surface diffusion model. In their suggestion, the cohesive normal force (Fn,ij) between two particles i and j is related to the bridge radius between them as:

24

𝐹𝑛,𝑖𝑗 = 𝜋𝑥 2 𝛿

(17)

Since x is time dependent, this kind of inter-particle cohesive force is a function of time. Mansourpour et al. (Mansourpour et al., 2014a) used this kind of forces for investigating the behavior of polyethylene powder in a fluidized bed under viscous sintering conditions. Another model which was introduced by Jagota and Scherer(Jagota and Scherer, 1993) predicts the normal cohesive contact force according to: 𝐹𝑛,𝑖𝑗 = 3𝜋𝑥𝜎𝑓(𝑥⁄𝑅 ) − 6𝜋µ𝑥

𝑅 𝑽. 𝒏 𝑑

(18)

where f(x/R) is a dimensionless function of bridge radius on the order of unity. The first term in Eq. 19 represents the cohesive force due to capillarity where the second term indicates viscous dissipation force. As showed in Table 3, recently a new model has been developed for calculating inter-particle sintering force based on particle scale results. This model suggests sintering force for first stages of sintering (0.1 ≤ 𝑥/𝑅0 ≤ 0.7) as (Katsura et al., 2015; Wakai et al., 2016): 𝑥 2 𝑥 𝐹𝑛,𝑖𝑗 = 𝜎𝑅0 [−11.53 ( ) + 13.58 ( ) + 0.4] 𝑅 𝑅

(19)

However, using this type of forces, no DEM simulations have been yet applied for predicting caking under viscous sintering conditions but it seems appropriate for such applications (Wakai et al., 2016). This is the only DEM inter-particle model which has been specified for viscous sintering.

7- Future Prospective With respect to all explained topics, futuristic discussions which will make our understanding from viscous sintering phenomenon more accurate are suggested briefly: In micro scale:  Ascertaining exact equation for bridge growth during Newtonian viscous sintering  Proposing inter-particle force models based on micro scale studies to be applied in macro scale modeling  Developing models considering thermal effects on particles properties (especially viscosity), non-spherical shapes of particles and porosity of particles  Determining obviously the effect of viscoelasticity on pair particle coalescence in all of its stages by considering different non-Newtonian fluid models In meso scale:  Developing spatio-temporal models to find clearly how space interference of surrounding particles affects sintering kinetics 25

In macro scale:  Performing more DEM simulations on viscous caking and determining how inter-particle forces become accurate in predicting large scale behavior Finally, a comprehensive multi-scale modeling can provide the opportunity for accurate understanding of caking of amorphous materials.

8- Conclusion Caking may lead to severe problems during handling, processing and utilization of powders since it could affect their physical, chemical, mechanical and end-use properties. The main mechanisms of cake formation and underlying conditions leading to caking are basically different in crystalline and amorphous materials. The former occurs due to dissolution/recrystallization cycles whereas the latter is caused mainly by viscous sintering. Experimental, modeling and simulation attempts have been conducted for better understanding of conditions leading to cake formation and to find the ways to compensate its adverse effects. These attempts have been mostly carried out in micro (inter-particle), meso (multi particles) and macro (bulk) scales due to multi-scale nature of caking phenomenon. Problem of amorphous powders caking caused by viscous flow sintering mechanism has been reviewed with emphasis on basics and theoretical aspects. This selection was based on the importance of the subject and the lack of comprehensive resources on this field. Analytical methods and numerical simulations are valuable predictable tools which the right use of them may reduce additional costs during particle processing significantly. Soared costs attached to caked powders processing in particular, modeling becomes even more pronounced. A series of sequential events happen until the cake is seen in a visible scale. These events are spread on a vast length and time spectrum which leads to a multi-scale problem. In modeling also, multiscaling must be addressed; the issue which has been neglected in most of the works. However, analytical methods were pioneer in the recognition of scientific society from the caking in meso scale where the theory established by Frenkel in 1945 opened a way which for many years has been followed by others. The model postulates that capillary forces are responsible for some kind of solid/amorphous bridges between particles because of intrinsic tendency of the surfaces to attain the minimum possible area. It postulates also that viscous forces are the main repulsive factors -of course in this small scale- acting against coalescence. Therefore, the kinetic of coalescence as a dynamic event would be the outcome of this competition between surface tension and viscous forces. Act of kinetic energy in this competition is neglected which causes anyway a defect in the predictions of the model. Analytical solutions were very helpful in explaining trends but they were entirely limited by their assumptions. Numerical methods are also facing some adversities during simulating inter-particle coalescence including singularity problem, moving boundary issue (in one phase simulations) or high viscosity/density ratio issue (in two-phase simulations) and high computational costs. Not concerning these difficulties, one can be hopeful that more details of the particle coalescence will 26

be revealed by using CFD. 3-D considerations, non-spherical particle shapes, unequal particles, concurrent multi particles contacts, non-isothermal conditions and non-Newtonian fluid assumption along with considering momentum conservation law instead of constant strain rate assumption were among the most outstanding features of numerical methods in comparison with analytical approaches. Despite of such innovations, large parts of numerical simulations were dedicated to approve or reject other’s findings so that tremendous variations in the results have been seen occasionally. These variations were mostly because of considering different initial bridge radius to particle radius ratios (as the initial configuration of the system), comparing the results of various regimes together, numerical errors along with methods and the discrepancies of modeled geometries from what occurs in reality. For example, in most cases the particles are assumed as a perfect sphere without porosity and with a smooth surface. Variation of these assumptions from real nature of particles is clear. Analytical methods could not be seriously applied for large scale description of powders caking so numerical methods are of great interest. Among numerical methods, DEM is the most relevant, more because of its discrete essence which makes it consistent with the powders nature. Lack of specialized inter-particle cohesive forces which are needed for accurate DEM simulations, the application of this method was limited in particular for viscous flow caking. This is despite of this fact that various inter-particle models have been developed for investigating of caking undergoing other mechanisms such as diffusion. However a significant gap is seen.

Acknowledgment This work has been supported by the Center for International Scientific Studies & Collaboration (CISSC) and French Embassy in Iran. Iran's National Elites Foundation (INEF) is also acknowledged for their valuable support (INEF-Grant No. BN096).

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Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S., Jan, Y.-J., 2001. A front-tracking method for the computations of multiphase flow. Journal of Computational Physics 169, 708-759. Tsantilis, S., Briesen, H., Pratsinis, S., 2001. Sintering time for silica particle growth. Aerosol Science & Technology 34, 237-246. Tsuji, Y., Tanaka, T., Ishida, T., 1992. Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder technology 71, 239-250. Utili, S., Nova, R., 2008. DEM analysis of bonded granular geomaterials. International Journal for Numerical and Analytical Methods in Geomechanics 32, 1997-2031. Van de Vorst, G., 1994. Numerical simulation of axisymmetric viscous sintering. Engineering analysis with boundary elements 14, 193-207. Van de Vorst, G., Mattheij, R.M., Kuiken, H.K., 1992. A boundary element solution for two-dimensional viscous sintering. Journal of Computational Physics 100, 50-63. Van de Vorst, G., Mattheij, R.M.M., 1992. Numerical Analysis of a 2-D viscous sintering problem with non-smooth boundaries. Computing 49, 239-263. Van der Hoef, M., Ye, M., van Sint Annaland, M., Andrews, A., Sundaresan, S., Kuipers, J., 2006. Multiscale modeling of gas-fluidized beds. Advances in chemical engineering 31, 65-149. Varnik, F., Rios, A., Gross, M., Steinbach, I., 2013. Simulation of viscous sintering using the lattice Boltzmann method. Modelling and Simulation in Materials Science and Engineering 21, 025003. Wakai, F., Katsura, K., Kanchika, S., Shinoda, Y., Akatsu, T., Shinagawa, K., 2016. Sintering force behind the viscous sintering of two particles. Acta Materialia 109, 292-299. Walker, G., Magee, T., Holland, C., Ahmad, M., Fox, N., Moffatt, N., 1997. Compression testing of granular NPK fertilizers. Nutrient Cycling in Agroecosystems 48, 231-234. Walker, G.M., Magee, T.R.A., Holland, C.R., Ahmad, M.N., Fox, J.N., Moffatt, N.A., Kells, A.G., 1998. Caking processes in granular NPK fertilizer. Industrial & engineering chemistry research 37, 435-438. Wang, H., Zhu, X., Liao, Q., Sui, P., 2010. Numerical simulation on coalescence between a pair of drops on homogeneous horizontal surface with volume-of-fluid method. Journal of superconductivity and novel magnetism 23, 1137-1140. Weber, M.W., Hrenya, C.M., 2006. Square-well model for cohesion in fluidized beds. Chemical Engineering Science 61, 4511-4527. Weuster, A., Brendel, L., Wolf, D., 2013. Simulation of sheared, caking powder, POWDERS AND GRAINS 2013: Proceedings of the 7th International Conference on Micromechanics of Granular Media. AIP Publishing, pp. 515-518. Williams, M.L., Landel, R.F., Ferry, J.D., 1955. The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. Journal of the American Chemical society 77, 3701-3707. Wonisch, A., Guillon, O., Kraft, T., Moseler, M., Riedel, H., Rödel, J., 2007. Stress-induced anisotropy of sintering alumina: Discrete element modelling and experiments. Acta Materialia 55, 5187-5199. Wonisch, A., Kraft, T., Moseler, M., Riedel, H., 2009. Effect of Different Particle Size Distributions on Solid‐State Sintering: A Microscopic Simulation Approach. Journal of the American Ceramic Society 92, 1428-1434. Xiong, Y., Pratsinis, S.E., 1993. Formation of agglomerate particles by coagulation and sintering—Part I. A two-dimensional solution of the population balance equation. Journal of Aerosol Science 24, 283-300. Yao, W., Maris, H., Pennington, P., Seidel, G., 2005. Coalescence of viscous liquid drops. Physical Review E 71, 016309. Zhang, R., Li, J., 2006. Simulation on mechanical behavior of cohesive soil by Distinct Element Method. Journal of Terramechanics 43, 303-316.

34

Zhou, H., Derby, J.J., 1998. Three‐Dimensional Finite‐Element Analysis of Viscous Sintering. Journal of the American Ceramic Society 81, 533-540. Zhou, H., Derby, J.J., 2001. An assessment of a parallel, finite element method for three‐dimensional, moving‐boundary flows driven by capillarity for simulation of viscous sintering. International Journal for Numerical Methods in Fluids 36, 841-865.

35

Figure 1- Stages of caking as a multi scale phenomenon

• Figure 2- Bridge formation in viscous sintering of two particles; initial contact (left), bridge has been formed (right). 36

190

Melting curve Solubility curve Tg (µ = 1.1×1014 µ = 1×1010 Pa.s µ = 1×1012 Pa.s

Temperature (C)

140 90

Solution (Emulsion)

40

Caking zone (Rubbery state) Lactose crystalls

-10

Glassy Tg solid state

Ice and solution -60 0

0.2

0.4

0.6

0.8

1

Weight Fraction of Lactose

Figure 3- Phase diagram of lactose-water two phase system; data are taken from (Haque and Roos, 2004; Paterson et al., 2015; Roos, 2009)

37

• Figure 4- Schematic of bridge formation between two particles; Initial shape (dashed curve), real shape after bridge formation (blue curve), prediction of the shape by Frenkel’s model (black curve). Gray area indicates the mass which is overpredicted by Frenkel’s model

• Figure 5- Hydrodynamic regimes in two particles coalescence

38

• Figure 6- Phase diagram of two particle coalescence (Paulsen et al., 2012).

39

Elastic contact Viscoelastic contact

x/R0

Viscous flow contact

Stretching contact dominated

Zipping contact dominated

t0

tvis

t

• Figure 7- Map regime of coalescence by considering softness of particles (Lin et al., 2001)

40

• Figure 8- Applications of discrete numerical methods for simulating particulate flows along with their time and length scales including quantum mechanics (QM), molecular dynamics (MD), dissipative particle dynamics (DPD), Brownian dynamics (BD) and discrete element method (DEM) (Li et al., 2011)

41

Table 4- Flow characteristic parameters at two different regimes of coalescence

Regime Stokesian Eulerian

Characteristic velocity Characteristic time 𝑅0 µ 𝑈𝑣𝑖𝑠𝑐 = 𝜎⁄µ 𝑡𝑣𝑖𝑠𝑐 = 𝜎 3 𝜎 1/2 𝜌𝑅 𝑈𝑖𝑛𝑒𝑟𝑡 = ( ) 𝑡𝑖𝑛𝑒𝑟𝑡 = ( )1/2 𝜌𝑅 𝜎

42

Consequence µ𝑈𝑣𝑖𝑠𝑐 𝐶𝑎 = =1 𝜎 2 𝜌𝑈𝑖𝑛𝑒𝑟𝑡 𝑅 𝑊𝑒 = =1 𝜎

Table 5-Models and analytical solutions for inter-particle bridge growth

Reference

Assumptions and features

(Frenkel, 1946)

For Newtonian viscous flow, Solving energy balance for two identical spheres (3D), Constant radius of particles, Small dihedral angle, Assuming constant strain rate

Eshelby (Shaler, 1949)

(Rumpf, 1958)

(Lontz, 1964)

(Cosgrove et al., 1976)

(Hopper, 1984)

(Pokluda et al., 1997)

(Bellehumeur et al., 1998)

Modification of Frenkel’s in order to satisfy the mass conservation equation, Assuming elongational flow and constant strain rate The same as the Frenkel’s model with considering external forces Modification of Frenkel’s model by considering a simple relaxation process which considers viscosity decreasing exponentially with time Approximate analytical solution for linear arrays of rods, Assuming the actual interface as locally planar (2D) Two identical cylinders (2D), For Newtonian viscous flow, Analytical model but numerical integration for solution Modification of Frenkel’s model by eliminating constant radius and small dihedral angle assumptions due to be valid in all stages of sintering, Using Runge-Kutta-Fehlberg numerical integration for solution Using Frenkel’s approach along with Pokluda’s expression for particle radius but for UCM1 fluid to consider viscoelasticity, Steady state stresses assumption

Main results and notes

𝑥 2 3 𝑡 ( ) = R0 2 𝑡𝑣𝑖𝑠𝑐

*Valid only in early stages of sintering (because of assumption of small dihedral angle) *Not satisfaction of conservation of mass (because of assumption of constant radius of particles) 𝑥 2 𝑡 ( ) = R 𝑡𝑣𝑖𝑠𝑐 *Referred as Frenkel-Eshelby model *Valid for small dihedral angle 𝑥 8 𝑡 2𝐹𝑡 t ( )2 = + R0 5 𝑡𝑣𝑖𝑠𝑐 5𝜋. 𝑅2 𝜇 *With the same limitations of Frenkel’s model 𝑥 2 3 𝑡 1 ( ) = [ ]2 R0 2 𝑡𝑣𝑖𝑠𝑐 1 − 𝑒 (−𝑡⁄𝜆)

𝑥 𝑥 2𝑡 ( ) + ln (1 − ) = − 𝑅 𝑅 𝑡𝑣𝑖𝑠𝑐 *Prediction for late stage of sintering *Not an explicit solution (plot of x/R vs t/t visc which suggests 𝑥 ∝ 𝑡. 𝑙𝑛𝑡) 𝑅 *Limited to two-dimensional and Newtonian flow 𝑑𝜃 2−5/3 cos(𝜃) sin(𝜃) [2 − cos(𝜃)]1/3 = 𝑑𝑡 𝑡𝑣𝑖𝑠𝑐 [1 − cos(𝜃)][1 + cos(𝜃)] *Not an explicit solution (plot of x/R0 vs t/t visc )

Model:

𝑑𝜃 2 𝑡𝑣𝑖𝑠𝑐 𝐾12 𝑑𝜃 ) + (2𝜆𝑅 𝐾1 + ) −1 =0 𝑑𝑡 𝐾2 𝑑𝑡 2−5/3 cos(𝜃) sin(𝜃) 𝐾2 = [2 − cos(𝜃)]5/3 [1 + cos(𝜃)]4/3 sin(𝜃) 𝐾1 = (1 + cos(𝜃))(2 − cos(𝜃))

8 (𝑅𝜆 𝐾1

Solution for initial stage of sintering:

1

Upper-Convected Maxwell

43

𝑥 𝑑( )2 −𝑅𝜆 − 2𝑡𝑣𝑖𝑠𝑐 + [(3𝑅𝜆)2 + 4𝑅𝜆𝑡𝑣𝑖𝑠𝑐 + 4𝑡𝑣𝑖𝑠𝑐 2 ]0.5 𝑅 = 𝑑𝑡 2(𝑅𝜆)2 *Not an explicit solution (plot of x/R0 as function of t/t visc and λ which shows that coalescence rate decreases with increasing relaxation time) *Good agreement with their experiments for low Deborah numbers only by considering unrealistic large values for relaxation time (Eggers 1999)

et

al.,

Solving the representation of equation in 2-D

integral Stokes

−𝐶𝑣𝑖𝑠𝑐 (

𝑥 = R0 {

(Scribben et al., 2006)

(Eggersdorfer et al., 2011)

Minor modification of Pokluda’s model, Modification of Bellehumeur’s model for considering a general viscoelastic constitutive model (UCM) at transient mode, Assuming elongational flow

Improvement Pokluda’s model to be appropriate for two particles with different sizes and for irregular multi-particle sintering

𝑡

𝑡 ) ln ( ) ∶ 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡 𝑡𝑣𝑖𝑠𝑐 𝑡𝑣𝑖𝑠𝑐

−𝐶𝑖𝑛𝑒𝑟𝑡 (

𝑡 𝑡𝑖𝑛𝑒𝑟𝑡

1⁄ 2

)

∶ 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡

*Demonstrating the equality of 2-D and 3-D analysis for initial stages of coalescence which lead to the same models but with different constants Newtonian model: 𝑑𝜃 𝜎 𝐾2 = 𝑑𝑡 𝑅𝜇 𝐾12 tan(𝜃) sin(𝜃) 2(2 − cos(𝜃)) + (1 + cos(𝜃)) − [ ] (1 + cos(𝜃))(2 − cos(𝜃)) 2 6 K2 was the same as (Bellehumeur et al., 1998) General coalescence model for viscoelastic materials 2 2 ⁄3 𝑅𝐾1 (𝜏𝑥𝑥 − 𝜏𝑦𝑦 ) − 1 = 0 3𝜎𝐾2 *Solving the model numerically See Eq.8 of their paper for mass balance model See Eq.14a of their paper for energy balance model 𝐾1 =

Bellehumeur, C.T., Kontopoulou, M., Vlachopoulos, J., 1998. The role of viscoelasticity in polymer sintering. Rheologica acta 37, 270-278. Cosgrove, G., Strozier Jr, J., Seigle, L., 1976. An approximate analytical model for the late‐stage sintering of an array of rods by viscous flow. Journal of Applied Physics 47, 1258-1264. Eggers, J., Lister, J.R., Stone, H.A., 1999. Coalescence of liquid drops. Journal of Fluid Mechanics 401, 293-310. Eggersdorfer, M.L., Kadau, D., Herrmann, H.J., Pratsinis, S.E., 2011. Multiparticle sintering dynamics: from fractal-like aggregates to compact structures. Langmuir 27, 6358-6367. Frenkel, J., 1946. Viscous flow of crystalline bodies. Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 16, 29-38. Hopper, R.W., 1984. Coalescence of two equal cylinders: exact results for creeping viscous plane flow driven by capillarity. Journal of the American Ceramic Society 67, C‐262-C‐264. Lontz, J.F., 1964. Sintering of polymer materials, Sintering and Plastic Deformation. Springer, pp. 25-47. 44

Pokluda, O., Bellehumeur, C.T., Vlachopoulos, J., 1997. Modification of Frenkel's model for sintering. AIChE journal 43, 3253-3256. Rumpf, H., 1958. Grundlagen und methoden des granulierens. Chemie Ingenieur Technik 30, 144-158. Scribben, E., Baird, D., Wapperom, P., 2006. The role of transient rheology in polymeric sintering. Rheologica acta 45, 825-839. Shaler, A., 1949. Seminar on the kinetics of sintering. Transactions of the American Institute of Mining and Metallurgical Engineers 185, 796-813.

45

Table 6- Numerical studies of particle coalescence

Reference

Assumptions features

(Ross et al., 1981)

2D, 𝑥

𝑛

( ) = 𝐹(𝑇)𝑡 𝑅0

and

main Numerical Main Results method (Interface treatment)

Assuming and trying to

find 𝑛 Starting from x/a = 0.16 as the initial configuration of the particles

(Hiram 1983)

and

FEM (first applying FEM to the micro scale problem of viscous sintering)

Nir,

2D, Two equal circles, Evaluating the effect of outer phase by changing viscosity ratios between inner and outer phases, Starting from a predefined initial condition with at least x/a > 0.1

Boundary method

(Van de Vorst et al., 1992)

Developing a boundary element method for simulating ِ2D surface driven sintering, Evaluating four different examples of arbitrary contact shapes between two particles 2D, Developing a special algorithm for node redistribution which is claimed to be suitable for the problems by curvature based driving force 2D (planar), Coalescence of two circular particles with the same and different radiuses

BEM2 based on linear or quadratic boundary elements

(Van de Vorst and Mattheij, 1992)

(Martínez‐ Herrera and Derby, 1994)

(Van de Vorst, 1994)

(Garabedian Helble, 2001)

and

(Kirchhof

al.,

2

et

2D with axisymmetric assumption (somehow 3D), Evaluating different contact shapes containing two spheres, two rings, one sphere on a ring and two cones 2-D (planar), Coalescence of two equal particles 3D, Studying bridge growth

integral

1 < 𝑛 < 2 𝑓𝑜𝑟 0.16 < < 0.6,

𝑛 > 2 𝑓𝑜𝑟 0.6 <

𝑥 𝑅0

𝑥

𝑅0

Finding that Frenkel’s model underpredicts the bridge growth, Confirming that Frenkel’s model is not appropriate for the whole stages of sintering. 𝑡 𝑥 = 𝑥𝑓 (1 − exp (− )) 𝑡𝑣𝑖𝑠𝑐 Finding similar behavior of interface evolution for a wide viscosity ratios (no serious effect was seen for outer phase), Good agreement with experimental results for polystyrene doublets Not a quantitative but qualitative results which confrim the applicability of their BEM algorithm in solving viscous sintering.

BEM based on linear or quadratic boundary elements

Validation of the method through agreement with the results of Hopper’s model (HOPPER, 1990).

FEM (with Monge projection representation of interface(, Front tracking method with algebraic mesh generation BEM

Good agreement with the results of Hopper’s model (HOPPER, 1990) for two identical particles, Finding that densification rate increased by increasing the ratio of two particles radiuses (𝑅1,0 /𝑅2,0 ). Finding discrepancies between 2-D and axisymmetric results, Representing results for bridge growth between different contact shapes

Boundary method

Good agreement with results in literature (Martínez‐ Herrera and Derby, 1994) Finding faster kinetics for larger

integral

Fractional VOF

Boundary Element Method

46

2009)

between two different sizes

(Djohari et al., 2009)

Comparison between viscous flow mechanism and vacancy diffusion mechanism

FEM

(Wang et al., 2010)

2-D, Coalescence of two drops on a horizontal surface Considering viscous flow and van der Waals forces coupled together for simulating nanoparticles sintering Investigating two-particle and many-particle systems

FVM3 (VOF)

(Gross et al., 2013)

2D and 3D

LBM

(Katsura et al., 2015)

Simulating shape changes of axisymmetric ellipsoidal particle as the later stage of sintering between two particles 2D, Coalescence of two cylinders

FEM

(Kirchhof 2012)

et

al.,

(Varnik et al., 2013)

(Wakai et al., 2016)

3

spheres

with

size differences of particles (agreement with results in literature (Martínez‐ Herrera and Derby, 1994))

Fractional VOF

LBM

FEM

Finite Volume Method

47

Finding that viscous flow leads to instantaneous shrinkage while diffusion mechanism shrinkage occurs with delay Good agreement with their experimental results Good agreement of the results with experiments Confirming Frenkel’s equation for early stage of viscous sintering, Predicting relative density and effective viscosity for many-particle case Verifying 𝑥 ∝ 𝑡 1/2 law (Frenkel’s law) Extracting inter-particle sintering force from neck growth results Verifying that sintering force as 𝐹𝑠 = −𝑃̅𝑎 + 𝜎𝑙 can predict 2-D sintering process well, Giving a force inter-particle model for viscous sintering (Eq.20)

Table 4- Models and solutions for multi-particle viscous sintering

Reference

Assumptions features

(Ross et al., 1981)

Simulating infinite line of identical cylinders (2D circles), Using FEM for numerical solution Simulating infinite line of identical cylinders (2D circles), Using FEM for numerical solution 2D, Using BEM for numerical solution Simulating two-particle system and system of infinite line of identical cylinders (2D circles) Using FEM for numerical solution Simulating coalescence of three linear particles, 2D with axisymmetric assumption (somehow 3D), Using FEM for numerical solution Investigating non-isothermal sintering of a column of particles by coupling conductive heat transfer with Pokluda’s model of sintering rate 3-D, Investigation for threeparticle (L shape) system, Using a parallel FEM code for numerical solution

(Jagota and Dawson, 1988)

(Van de Vorst and Mattheij, 1992) (Martínez‐ Herrera and Derby, 1994)

(Martínez‐ Herrera and Derby, 1995)

(Bergman, 1997; Tarafdar and Bergman, 2002)

(Zhou and Derby, 1998)

and Main results and remarks

(Kirchhof et al., 2009)

Studying bridge growth between particles in some various agglomerate morphologies with total nine initial contacts (ten particles)

(Eggersdorfer al., 2011)

Investigating the evolution of the morphology, radius of gyration and effective fractal dimension of ensembles of irregular particles, Numerical solution of the model by SHAKE algorithm

et

Predicting bridge growth rate faster than results of fixedgeometry models such as Frenkel’s model

Predicting bridge growth rate slower than results of Frenkel’s model

Qualitative results for systems with three particles Coalescence was seen faster in infinite line of particles in comparison with two-particle case

Confirming the independency of different contacts from each other only at first stages of sintering (This matter was suggested before (Jagota and Dawson, 1990) Finding the overall shrinkage of three-particle system faster than two-particle one. Predicting physical and thermal responses to various thermal and initial conditions and also to different sizes of the powder

Validation by the literature (Martínez‐ Herrera and Derby, 1994) (axisymmetric), Revealing some effects in 3-particle system not predictable by 2-partices case, Observing anisotropic shrinkage and overall inward rotation of outer particles in 3-particle case, Demonstrating the importance of 3-D simulations Deriving a relation for describing first stage of sintering as a function of average coordination number ̅̅ 𝑁̅̅𝑘 : 𝑆 𝑑( ⁄𝑆 ) ̅̅ 𝑁̅̅𝑘 𝑡 0 = 𝐾0 𝑑𝑡 𝑡𝑣𝑖𝑠𝑐 Validating the method Proposing expressions for the evolution of fractal dimension and the surface area of aggregates undergoing viscous sintering

48